PRAMANA
c Indian Academy of Sciences
— journal of
physics
Vol. 85, No. 2
August 2015
pp. 281–289
Fission barrier heights in the A ∼ 200 mass region
K MAHATA
Nuclear Physics Division, Bhabha Atomic Research Centre, Mumbai 400 085, India
E-mail: [email protected]
DOI: 10.1007/s12043-015-1042-4; ePublication: 19 July 2015
Abstract. Statistical model analysis is carried out for p- and α-induced fission reactions using a
consistent description for fission barrier and level density in A ∼ 200 mass region. A continuous
damping of shell correction with excitation energy is considered. Extracted fission barriers agree
well with the recent microscopic–macroscopic model. The shell corrections at the saddle point were
found to be insignificant.
Keywords. Fission barrier; damping of shell correction; statistical model.
PACS Nos 25.70.Jj; 24.10.Pa; 24.75.+i; 27.80.+w
1. Introduction
Experimental determination of fission barrier height in the mass region A ∼ 200 continues to be a challenging problem. Accurate knowledge of fission barrier height is vital
not only to understand the heavy-ion-induced fusion–fission dynamics and prediction of
superheavy elements, but also other areas, such as stellar nucleosynthesis and nuclear
energy applications. In the actinide region, the fission barrier heights are comparable to
the neutron separation energies and could be determined accurately from the measured
fission excitation functions, which exhibit a characteristic rise in the barrier energy followed by a flat plateau. In the A ∼ 200 mass region, fission barrier heights are much
higher than the neutron separation energies. Most of the measurement of fission crosssections in this mass region are performed at energies much higher than the fission barrier,
where there are other open channels and a statistical description is essential.
Although a number of studies have been carried out, there are still ambiguities in
choosing various input parameters for the statistical model analysis. According to the
statistical model of compound nucleus (CN) decay, the probabilities of decay to different
channels are governed by the transmission coefficient and relative density of states (phasespace). The nuclear level density depends on the level density parameter (a) related to
the single-particle density near the Fermi surface and the available thermal energy (U ).
The ground-state shell corrections in the nuclei around the doubly closed shell nucleus
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281
K Mahata
208
Pb (Z = 82, N = 126) are large and its damping with excitation energy has to be
incorporated accurately in the statistical model analysis. The nuclear level density of a
shell-closed nucleus shows the same energy dependence at high excitation energy (>40
MeV) as that of nuclei away from shell closure, if the excitation energy is measured
from the liquid-drop surface, indicating the complete washing out of the shell corrections
at those energies [1]. At intermediate energy, the dependence is phenomenologically
described in terms of energy-dependent level density parameter approaching asymptotically to the liquid-drop value [2]. The phenomenological description of gradual damping
of the ground-state shell correction with excitation energy was obtained by examining the
density of neutron resonances situated near the neutron threshold (∼8 MeV). Recently, it
has also been studied by measuring evaporation spectra in the 208 Pb region [3]. The knowledge about the shell corrections at the saddle point in A ∼ 200 mass region is ambiguous.
The heavy-ion-induced fission excitation functions are not sensitive to the correlated
variation of the fission barrier height and the ratio of the level density parameter at the saddle point to that at the equilibrium deformation (ãf /ãn ) [4–7]. However, the pre-fission
neutron multiplicity (νpre ) data are sensitive to this correlated variation and hence it can be
used to constrain the statistical model parameters. However, the measured νpre can have
dynamical contribution, which should be taken care of. Analysis [7] of the fission and
evaporation residue cross-sections along with pre-fission neutron multiplicity data for the
12
C+198 Pt system required large shell corrections at the saddle point and yielded fission barriers much smaller (13.4 MeV) than those (∼22 MeV) obtained for the same
compound nuclei from the analysis of light-ion-induced reactions. In the analysis, the
νpre data were corrected for a dynamical emission corresponding to the fission delay of
30×10−21 s.
In this article, we discuss the statistical model analysis for p+209 Bi, α+184 W, 206,208 Pb,
209
Bi systems using the same prescription for level density and fission barrier as in [7].
The experimental fission excitation functions are taken from [8–12].
2. Statistical model analysis
The statistical model analyses have been carried out using the code PACE [13] with a modified prescription for fission barrier and level density [7]. The fission barrier is expressed
as
Bf (J ) = cf × BFRFRM (J ) − n + f ,
(1)
where cf , n and f are the scaling factors to the rotating finite-range model (RFRM)
fission barrier [14], shell correction at the equilibrium and shell correction at the saddle
deformation, respectively. The shell corrections at the equilibrium deformations are taken
from [15]. Fermi gas level density formula was used to calculate the level densities at
the equilibrium and saddle-point deformations. The excitation energy of the compound
nucleus (CN) is calculated as
Un = Ec.m. + Q − Erot (J ) − δp ,
(2)
where Ec.m. , Q, Erot (J) and δp are the energy in the centre-of-mass (c.m.), Q-value for
fusion, rotational energy and pairing energy, respectively. The Q-value for fusion as
well as the particle separation energies for subsequent decays are calculated using the
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Fission barrier heights in A ∼ 200 mass region
experimental masses [16]. The excitation energy available at the saddle-point deformation
is taken as Uf = Un − Bf (J ). The damping of the shell correction with excitation energy
is realized by assuming energy-dependent level density parameter as
ax (U ) = ãx [1 + (x /Ux )(1 − e−ηUx )]
(3)
with x = n and f corresponding to the equilibrium deformation and saddle-point deformation, respectively. The asymptotic liquid-drop value of the level density parameter at
the equilibrium deformation is taken as ãn = A/9 MeV−1 . The asymptotic liquid-drop
value of the level density parameter at the saddle-point deformation (ãf ) may be different
from that of ãn due to the difference in nuclear shapes at these two cases. Hence, the ratio
ãf /ãn was kept as a free parameter to be decided by the fit to the data.
The spin distribution of CN is taken as
σ (ℓ) =
k 2 [1
π(2ℓ + 1)
.
+ exp((ℓ − ℓmax )/δℓ)]
(4)
The value of ℓmax and δℓ are determined from the experimental fusion cross-section (σfus )
and fission fragment angular anisotropy data. The sum of the xn cross-sections available
in [17] was taken as fusion cross-section for the p+209 Bi system. For α+206 Pb system, xn
cross-sections are not available. However, xn cross-sections for the α+209 Bi system are
available [18]. Statistical model analysis of α+209 Bi system considering fusion crosssection from the Bass systematics [19] reproduces the experimental xn cross-sections
well. Hence, fusion cross-sections for α-induced reactions were estimated using the Bass
systematics. The values of δℓ = 2 and 3 reproduce the experimental fission fragment
angular anisotropy data for p- and α-induced reactions, respectively.
In figures 1 and 2, the statistical model predictions are compared with the experimental
xn cross-sections for p- and α-induced reaction on 209 Bi target. Fusion process leads
to bell-shaped excitation functions for xn channels and the presence of pre-equilibrium
particle emission gives rise to high-energy tails to these distributions. A significant contribution from pre-equilibrium particle emission to the xn cross-sections will lead to
σ (mb)
103
102
2n
3n
4n
5n
101
25
30
35
E*
40
45
(MeV)
50
55
Figure 1. Statistical model predictions of xn cross-sections for the p+209 Bi system
are compared with the experimental data.
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283
K Mahata
σ (mb)
103
102
2n
3n
4n
5n
101
100
20
25
30
35
E*
40
45
50
55
(MeV)
Figure 2. Same as figure 1 for α+209 Bi system.
10-1
209 Bi
p+
10-2
206 Pb × 0.1
α+
10-3
Pf
10-4
10-5
10-6
10-7
10-8
20
25
30
35
40
45
50
55
60
65
E* (MeV)
Figure 3. Experimental fission probabilities for p+209 Bi and α+206 Pb systems are
compared with the statistical model calculations using parameters from [7] (Bf (0) =
13.4 MeV) obtained from the fit to the experimental ER and fission excitation functions
along with the pre-fission neutron multiplicity data for the 12 C+198 Pt system.
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Fission barrier heights in A ∼ 200 mass region
overestimation of the fusion cross-section. As can be seen from the figures, the distribution of the xn cross-sections could be well reproduced, except for the high-energy tails in
the experimental distributions. We have estimated the contribution of the pre-equilibrium
emission to xn cross-sections from these high-energy tails. It was found that the preequilibrium contribution to the xn cross-sections is of the order of 10% for excitation
energies below 40 MeV and it is of the order of 20% at 50 MeV for p-induced reaction.
Fission following pre-equilibrium particle emission will be negligible because of the population of the target-like nuclei with much lower excitation energies than that of the CN.
For the α-induced reaction, the pre-equilibrium contribution to xn cross-section is found
to be insignificant in the energy range considered in the present analysis.
In figure 3, the experimental fission probabilities (Pf = σfis /σfus ) for p+209 Bi and
α+206 Pb systems are compared with the predictions of the statistical model using the
parameters (Bf = 13.4 MeV) of [7], which fits the fission excitation functions and the
νpre data for 12 C+198 Pt system simultaneously. The statistical model calculation using the
parameters of [7] fails to reproduce the shape of the excitation functions. Attempts to fit
the excitation functions by varying the values of f and ãf /ãn result in f ∼ 0.5 MeV.
As the resulted shell correction at the saddle point (f ) is within the uncertainty of the
RFRM prediction and the damping of shell corrections at the saddle point is also not well
known, the value of f was assumed to be zero and the values of cf and ãf /ãn were
varied to fit the excitation functions.
Initially, we attempted to fit the excitation functions using an energy-independent
damping factor η = 0.054 MeV−1 [2] for damping of the shell correction at the equilibrium
deformation. It could not fit the excitation functions for the entire excitation energy range
2.5
p + 209Bi
α + 206Pb
α + 184W
Ratio
2
1.5
1
0.5
20
30
40
50
60
70
E*(MeV)
Figure 4. Ratio of the experimental fission probability to the statistical model prediction using energy-independent damping factor η = 0.054 MeV−1 . The lines are to
guide the eye.
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285
K Mahata
considered in the analysis. The ratio of the experimental fission probabilities to the predictions of the statistical model using energy-independent damping factor for p+209 Bi,
α+184 W, 206 Pb systems are shown in figure 4. While the ratio shows a systematic deviation
10-1
10-2
Pf
10-3
10-4
~a /a~ = 1.03
f n
~a /a~ = 1.05
f n
~a /a~ = 1.07
f n
10-5
10-6
25
30
35
40 45
E*(MeV)
50
55
60
Figure 5. Statistical model predictions of fission probabilities for the p+209 Bi system
with different values of ãf /ãn for a fixed value of Bf .
10-1
10-2
Pf
10-3
10-4
10-5
10-6
Bf = 21.3 MeV
Bf = 21.8 MeV
Bf = 22.2 MeV
25
30
35
40 45
E*(MeV)
50
55
60
Figure 6. Statistical model predictions of fission probabilities with different values of
Bf for the p+209 Bi system. The values of ãf /ãn are varied to get the best-fit.
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Fission barrier heights in A ∼ 200 mass region
from unity for the p+209 Bi, α+206 Pb (n = −10.6 MeV) systems, no such deviation
was observed for the α+184 W (n = −1.9 MeV) system. From this observation it was
concluded that the deviation is due to the improper damping of the shell correction with
excitation energy. An energy-dependent shell damping factor η = 0.054 + 0.002 × Un is
found to give better agreement with the experimental data.
We have also studied the sensitivity of the parameters to the excitation functions. Predictions of the statistical model with different values of ãf /ãn for Bf = 21.8 MeV are
compared with experimental fission probabilities for the p+209 Bi systems in figure 5. It
is found that the lower part of the excitation function is not sensitive to the values of
ãf /ãn . The predictions of the statistical model with different values of Bf for p+209 Bi
system are shown in figure 6. For each value of fission barrier, the value of ãf /ãn was
varied to obtain the best-fit. The low-energy part of the fission excitation functions are
found to be very sensitive to the variation of Bf . It should be mentioned here that preequilibrium particle emission will not have significant effect at the lower part of the excitation function and hence on the extracted fission barrier height.
The results of the statistical model calculations are shown in figure 7. Best-fit
parameters are listed in table 1.
100
209 Bi ×
α+
10
208 Pb ×
10
α+
10-1
209 Bi
p+
10-2
206 Pb × 0.1
α+
10-3
184 W
× 10
Pf
α+
10-4
10-5
10-6
10-7
10-8
20
25
30
35
40 45
E* (MeV)
50
55
60
65
Figure 7. Experimental fission probabilities for p- and α-induced reactions are
compared with the statistical model calculations using the parameters given in table 1.
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287
K Mahata
Table 1. Relevant statistical model parameters corresponding to best-fits. Fission
exp
barriers extracted from the present analysis (Bf (0)) are also compared with the
macroscopic–microscopic finite-range liquid-drop model (LDM) [20] fission barrier
(Bfcal ).
System
a˜f /a˜n
cf
n
(MeV)
α+209 Bi →213 At
α+208 Pb →212 Po
p+209 Bi →210 Po
α+206 Pb →210 Po
α+184 W →188 Os
1.034
1.035
1.042
1.051
1.082
1.04
1.06
1.04
1.07
1.18
−7.51
−8.45
−10.62
−10.62
−1.89
exp
Bf (0)
(MeV)
Bfcal [20]
17.8±0.2
20.3±0.3
21.8±0.2
22.1±0.3
24.3±0.6
18.56
20.27
22.14
22.14
–
3. Summary and conclusion
We have carried out the statistical model calculations for p- and α-induced reactions to
determine the fission barrier heights in the A ∼ 200 mass region. Sensitivity of the relevant parameters was studied. While the low-energy part of the excitation functions was
found to be sensitive to the height of the fission barrier, high-energy part of the excitation functions was found to be sensitive to the value of ãf /ãn . Effect of pre-equilibrium
particle emission on the extracted fission barrier was estimated to be insignificant. The
shell correction at the saddle point was also found to be insignificant. However, the
statistical model calculation without shell correction at the saddle point substantially
underpredicted the measured pre-fission neutron multiplicity data in heavy-ion fusion–
fission reactions [7,21]. Further investigation is required to study the statistical nature of
these pre-fission neutrons and contributions of other sources (e.g., dynamical emission
and near-scission emission).
Acknowledgement
The author gratefully acknowledges the suggestions and advice of S Kailas and S S
Kapoor.
References
[1] V S Ramamurthy, S S Kapoor and S K Kataria, Phys. Rev. Lett. 25, 386 (1970)
[2] A V Ignatyuk, G N Smirenkin and A S Tishin, Sov. J. Nucl. Phys. 21, 255 (1975), [Yad. Fiz.
21, 485 (1975)]
[3] P C Rout, D R Chakrabarty, V M Datar, Suresh Kumar, E T Mirgule, A Mitra, V Nanal, S P
Behera and V Singh, Phys. Rev. Lett. 110, 062501 (2013), DOI: 10.1103/PhysRevLett.110.
062501
[4] S E Vigdor, H J Karwowski, W W Jacobs, S Kailas, P Singh, F Foga and P Yip, Phys. Lett B
90, 384 (1980)
[5] D Ward, R J Charity, D J Hinde, J R Leigh and J O Newton, Nucl. Phys. A 403, 189 (1983)
[6] K Mahata, S Kailas, A Shrivastava, A Chatterjee, A Navin, P Singh, S Santra and B S Tomar,
Nucl. Phys. A 720, 209 (2003)
288
Pramana – J. Phys., Vol. 85, No. 2, August 2015
Fission barrier heights in A ∼ 200 mass region
[7] K Mahata, S Kailas and S S Kapoor, Phys. Rev. C 74, 041301(R) (2006)
[8] A Khodai-Joopari, Ph.D. thesis (University of California, 1966), Report UCRL-16489
[9] O A Zhukova, A V Ignatyuk, M G Itkis, S I Mulgin, V N Okolovich, G N Smirenkin and A S
Tishin, Sov. J. Nucl. Phys. 26, 251 (1977), [Yad. Fiz. 260, 473 (1977)]
[10] A V Ignatyuk, M G Itkis, I A Kamenev, S I Mulgin, V N Okolovich and G N Smirenkin, Sov.
J. Nucl. Phys. 40, 400 (1984), [Yad. Fiz. 40, 625 (1984)]
[11] L G Moretto, K X Jing, R Gatti, G J Wozniak and R P Schmitt, Phys. Rev. Lett. 75, 4186
(1995)
[12] A D’Arrigo, G Gardiana, M Herman, A V Ignatyuk and A Taccone, J. Phys. G: Nucl. Part 20,
365 (1994)
[13] A Gavron, Phys. Rev. C 21, 230 (1980)
[14] A J Sierk, Phys. Rev. C 33, 2039 (1986)
[15] W D Myers and W J Swiatecki, Lawrence Berkeley Laboratory Report No. LBL-36803, 1994
[16] M Wang, G Audi, A H Wapstra, F G Kondev, M MacCormick, X Xu and B Pfeiffer, Chin.
Phys. C 36, 1603 (2012)
[17] EXFOR database: http://www-nds.iaea.org/EXFOR/
[18] Yu E Penionzhkevich, Yu A Muzychka, S M Lukyanov, R Kalpakchieva, N K Skobelev, V P
Perelygin and Z Dlouhy, Eur. Phys. J. A - Hadrons Nuclei 13(1–2), 123 (2002), DOI: 10.1140/
epja1339-23
[19] R Bass, Phys. Rev. Lett. 39, 265 (1977)
[20] Peter Möller, Arnold J Sierk, Takatoshi Ichikawa, Akira Iwamoto, Ragnar Bengtsson, Henrik
Uhrenholt and Sven Åberg, Phys. Rev. C 79, 064304 (2009)
[21] K S Golda et al, Nucl. Phys. A 913, 157 (2013)
Pramana – J. Phys., Vol. 85, No. 2, August 2015
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